Lecture 2

Latent Gaussian Models and INLA

Sara Martino
sara.martino@ntnu.no

Dept. of Mathematical Science, NTNU

Outline

  • Latent Gaussian Models

Repetition

Everything in R-INLA is based on so-called latent Gaussian models

— A few hyperparameters \(\theta\sim\pi(\theta)\) control variances, range and so on

— Given these hyperparameters we have an underlying Gaussian distribution \(\mathbf{u}|\theta\sim\mathcal{N}(\mathbf{0},\mathbf{Q}^{-1}(\theta))\) that we cannot directly observe

— Instead we make indirect observations \(\mathbf{y}|\mathbf{u},\theta\sim\pi(\mathbf{y}|\mathbf{u},\theta)\) of the underlying latent Gaussian field

Repetition

Models of this kind: \[ \begin{aligned} \mathbf{y}|\mathbf{x},\theta &\sim \prod_i \pi(y_i|\eta_i,\theta)\\ \mathbf{\eta} & = A_1\mathbf{u}_1 + A_2\mathbf{u}_2+\dots + A_k\mathbf{u}_k\\ \mathbf{u},\theta &\sim \mathcal{N}(0,\mathbf{Q}(θ)^{−1})\\ \theta & \sim \pi(\theta) \end{aligned} \]

occurs in many, seemingly unrelated, statistical models.

Examples

  • Generalised linear (mixed) models
  • Stochastic volatility
  • Generalised additive (mixed) models
  • Measurement error models
  • Spline smoothing
  • Semiparametric regression
  • Space-varying (semiparametric) regression models
  • Disease mapping
  • Log-Gaussian Cox-processes
  • Model-based geostatistics (*)
  • Spatio-temporal models
  • Survival analysis
  • +++

Main Characteristics

  1. Latent Gaussian model
  2. The data are conditionally independent given the latent field
  3. The predictor is linear1
  4. The dimension of \(\mathbf{u}\) can be big (\(10^3-10^6\))
  5. The dimension of \(\theta\) should be not too big.